My non-inverting amplifier doesn’t work correctly (the Gain isn’t correct) [ICL7650S Op-Amp]

Your schematic is wrong. V2 should either be +11.75V or you should flip it.

The circuit (with +/-12V-ish supplies) should work. However, you've got an offset pot- it can adjust the input offset to +/-4mV typically. If you crank it all the way down, the input can appear to be 9mV - 4mV = 5mV. Nominal gain is about 350, so this still appears to be off. You might want to parallel R2 with a small capacitor (10nF ceramic, for example) in case the op-amp is oscillating (if you have an oscilloscope, look at the output directly). Leave out the offset pot to begin with- even the cheap version is within 150uV Vos without the pot. Bypass the supplies too, near the op-amp.

The OP-07 is an ancient 'precision' op-amp with a distinguished and storied past, however it's not a rail-to-rail input or output or even a single supply op-amp. It is possible to run it from +/-5V (with some care) but not (reliably) from 0/5V.

First we must start with some basic assumptions, we assume that the inputs of the circuit are fed from ideal voltage sources and that the op-amp is an ideal op-amp. The latter means that the circuit is linear and we can apply the principle of superposition to calculate the two gains..

For the inverting gain we treat things as if the + input of the circuit (not the + input of the op-amp) is connected to ground. The bottom two resistors have no affect since the op-amps inputs are high-impedance and the top two resistors act just like a normal inverting amplifier.

-Rf/Rg = -20/5 = -4

For the non-inverting gain we treat things as if the - input of the circuit (not the - input of the op-amp) is connected to ground.

When we do this we find that the non-inverting gain is made up of two stages. First we have a voltage divider formed by the bottom two resistors in the diagram. Then we have the op-amp along with the top two resistors in the diagram acting as a non-inverting amplifier. So the overall non-inverting gain works out to

(30/(20+30)) * (1+(20/5)) = (3/5) * 5 = 3